Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
Probabilistic checking of proofs and hardness of approximation problems
Probabilistic checking of proofs and hardness of approximation problems
A sieve algorithm for the shortest lattice vector problem
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Sampling Short Lattice Vectors and the Closest Lattice Vector Problem
CCC '02 Proceedings of the 17th IEEE Annual Conference on Computational Complexity
Sampling methods for shortest vectors, closest vectors and successive minima
Theoretical Computer Science
Proceedings of the forty-second ACM symposium on Theory of computing
Algorithms for the shortest and closest lattice vector problems
IWCC'11 Proceedings of the Third international conference on Coding and cryptology
A O(1/ε2)n-time sieving algorithm for approximate integer programming
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
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We provide the currently fastest randomized (1+epsilon)-approximation algorithm for the closest lattice vector problem in the infinity-norm. The running time of our method depends on the dimension n and the approximation guarantee epsilon by 2(O(n)) (log(1/epsilon))(O(n)) which improves upon the (2+1/epsilon)(O(n)) running time of the previously best algorithm by Blömer and Naewe. Our algorithm is based on a solution of the following geometric covering problem that is of interest of its own: Given epsilon0, how many ellipsoids are necessary to cover the scaled unit cube [-1+epsilon, 1-epsilon]n such all ellipsoids are contained in the standard unit cube [-1,1]n. We provide an almost optimal bound for the case where the ellipsoids are restricted to be axis-parallel. We then apply our covering scheme to a variation of this covering problem where one wants to cover the scaled cube with boxes that, if scaled by two, are still contained in the unit cube. Thereby, we obtain a method to boost any 2-approximation algorithm for closest-vector in the infinity-norm to a (1+epsilon)-approximation algorithm that has the desired running time.